1. David Kauchak cs302 Spring 2012
Sorting Conclusions
David Kauchak
cs302
Spring 2012

2. Administrative
Assignment 5 out today

3. Sorting bounds Mergsort is O(n log n)
Quicksort is O(n log n) on average
Can we do better?

4. Comparison-based sorting
Sorted order is determined based only on a comparison between input elements
A[i] < A[j]
A[i] > A[j]
A[i] = A[j]
A[i] ≤ A[j]
A[i] ≥ A[j]
Do any of the sorting algorithms we’ve looked at use additional information?

5. Comparison-based sorting
Sorted order is determined based only on a comparison between input elements
A[i] < A[j]
A[i] > A[j]
A[i] = A[j]
A[i] ≤ A[j]
A[i] ≥ A[j]
Do any of the sorting algorithms we’ve looked at use additional information? No
All the algorithms we’ve seen are comparison-based sorting algorithms

6. Comparison-based sorting
Sorted order is determined based only on a comparison between input elements
A[i] < A[j]
A[i] > A[j]
A[i] = A[j]
A[i] ≤ A[j]
A[i] ≥ A[j]
In Java (and many languages) for a class of objects to be sorted we define a comparator
What does it do?

7. Comparison-based sorting
Sorted order is determined based only on a comparison between input elements
A[i] < A[j]
A[i] > A[j]
A[i] = A[j]
A[i] ≤ A[j]
A[i] ≥ A[j]
In Java (and many languages) for a class of objects to be sorted we define a comparator
What does it do?
Just compares any two elements
Useful for comparison-based sorting algorithms

8. Comparison-based sorting
Sorted order is determined based only on a comparison between input elements
A[i] < A[j]
A[i] > A[j]
A[i] = A[j]
A[i] ≤ A[j]
A[i] ≥ A[j]
Can we do better than O(n log n) for comparison based sorting approaches?

9. Decision-tree model
Full binary tree representing the comparisons between elements by a sorting algorithm
Internal nodes contain indices to be compared
Leaves contain a complete permutation of the input
Tracing a path from root to leave gives the correct reordering/permutation of the input for an input
1:3 ≤
>
[3, 12, 7]
| 1,3,2 |
[3, 7, 12]
| 1,3,2 |
[7, 3, 12]
| 2,1,3 |
[3, 7, 12]

10. A decision tree model 1:2 ≤ > 1:3 2:3 ≤ > ≤ > |2,1,3| 1:3 2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|

11. A decision tree model [12, 7, 3] 1:2 ≤ > 1:3 2:3 ≤ > ≤ >
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[12, 7, 3]

12. A decision tree model [12, 7, 3] Is 12 ≤ 7 or is 12 > 7? 1:2 ≤ >
1:3
2:3 ≤
> ≤
>
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[12, 7, 3]
Is 12 ≤ 7 or is 12 > 7?

13. A decision tree model [12, 7, 3] Is 12 ≤ 3 or is 12 > 3? 1:2 ≤ >
1:3
2:3 ≤
> ≤
>
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[12, 7, 3]
Is 12 ≤ 3 or is 12 > 3?

14. A decision tree model [12, 7, 3] Is 12 ≤ 3 or is 12 > 3? 1:2 ≤ >
1:3
2:3 ≤
> ≤
>
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[12, 7, 3]
Is 12 ≤ 3 or is 12 > 3?

15. A decision tree model [12, 7, 3] Is 12 ≤ 3 or is 12 > 3? 1:2 ≤ >
1:3
2:3 ≤
> ≤
>
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[12, 7, 3]
Is 12 ≤ 3 or is 12 > 3?

16. A decision tree model [12, 7, 3] Is 7 ≤ 3 or is 7 > 3? 1:2 ≤ >
1:3
2:3 ≤
> ≤
>
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[12, 7, 3]
Is 7 ≤ 3 or is 7 > 3?

17. A decision tree model [12, 7, 3] Is 7 ≤ 3 or is 7 > 3? 1:2 ≤ >
1:3
2:3 ≤
> ≤
>
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[12, 7, 3]
Is 7 ≤ 3 or is 7 > 3?

18. A decision tree model 3, 2, 1 [12, 7, 3] 1:2 ≤ > 1:3 2:3 ≤ > ≤
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
3, 2, 1
[12, 7, 3]

19. A decision tree model 3, 2, 1 [3, 7, 12] [12, 7, 3] 1:2 ≤ > 1:3 2:3
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
3, 2, 1
[3, 7, 12]
[12, 7, 3]

20. A decision tree model [7, 12, 3] 1:2 ≤ > 1:3 2:3 ≤ > ≤ >
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[7, 12, 3]

21. A decision tree model [7, 12, 3] 1:2 ≤ > 1:3 2:3 ≤ > ≤ >
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[7, 12, 3]

22. A decision tree model [7, 12, 3] 1:2 ≤ > 1:3 2:3 ≤ > ≤ >
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[7, 12, 3]

23. A decision tree model [7, 12, 3] 1:2 ≤ > 1:3 2:3 ≤ > ≤ >
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[7, 12, 3]

24. A decision tree model [7, 12, 3] 1:2 ≤ > 1:3 2:3 ≤ > ≤ >
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[7, 12, 3]

25. A decision tree model [3, 7, 12] [7, 12, 3] 1:2 ≤ > 1:3 2:3 ≤ >
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|
[3, 7, 12]
[7, 12, 3]

26. How many leaves are in a decision tree?
Leaves must have all possible permutations of the input
What if decision tree model didn’t?
Input of size n, n! leaves
Some input would exist that didn’t have a correct reordering

27. A lower bound What is the worst-case number of comparisons for a tree?
1:2 ≤
>
1:3
2:3 ≤
> ≤
>
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|

28. A lower bound The longest path in the tree, i.e. the height 1:2 ≤ >
1:3
2:3 ≤
> ≤
>
|2,1,3|
1:3
2:3
|1,2,3| ≤
> ≤
>
|1,3,2|
|3,1,2|
|2,3,1|
|3,2,1|

29. A lower bound
What is the maximum number of leaves a binary tree of height h can have?
A complete binary tree has 2h leaves
log is monotonically increasing
from hw 

30. Can we do better?